Mutation graphs of asexual diploid organisms

Mutation graphs of asexual diploid organisms ´ Oscar J. Falc´ona, Ra´ul M. Falc´onb, Juan N´u˜neza (a) Departamento de Geometr´ıa y Topolog´ıa, Universidad de Sevilla. E-mail: [email protected], [email protected] (b) Departamento de Matem´atica Aplicada I, Universidad de Sevilla E-mail: [email protected] Abstract. The mutation graph of an asexual diploid organism is introduced as an edgecoloured graph derived from the genetic pattern by isotopisms of an evolution algebra over a finite field. We describe the step-by-step construction of this graph and establish some of its basic properties. In order to illustrate this construction, we focus on the spectrum of genetic patterns of two distinct genotypes during a mitosis process. Keywords. Line graph, evolution algebra, isomorphism. 1 Introduction In order to simulate algebraically self-reproduction processes in Genetics, Tian and Vojtechovsky [8, 9] introduced the concept of evolution algebra on a set β= {e1,...,en}of distinct genotypes with respect to a given phenotype of an asexual organism as an n-dimensional algebra over a field Khaving βas a natural basis such that eiej= 0, whenever i6=j. The tuple (e1e1,...,enen) is called the genetic pattern of the algebra. Evolution algebras present interesting connections with graph theory. In this regard, Tian [8] defined the evolution algebra related to a graph Gwith a finite set of vertices {v1,...,vn}as the algebra of basis {e1,...,en} described as e2 i=Pk∈Γ(vi)ekand eiej= 0, for all i, j ∈ {1,...,n}such that i6=j. Here, Γ(vi) denotes the set of neighbours of the vertex viin G. This definition enabled him to contemplate as a further work the development of known results on graph theory in the context of evolution algebras. More recently, Elduque and Labra [3, 4], and Cabrera et al. [1] dealt with the reverse problem. The former associated a weighted digraph to a given finite-dimensional evolution algebra and proved that the nonexistence of oriented cycles in such a graph is equivalent to the nilpotency of the corresponding algebra. The latter, in turn, described an alternative directed graph whatever the dimension of the evolution algebra is, from which its annihilator and irreducibility can be determined. Unlike Tian’s graphs, whose isomorphisms are equivalent to those of the evolution algebras Proceedings of X Encuentro Andaluz de Matemática Discreta, pp. 103-106. ISBN 978-84-697-4743-8. (La Línea, Cádiz, July 10-11, 2017). 2Mutation graphs of asexual diploid organisms under consideration, both proposals depend on the selected basis of the algebra so that isomorphic algebras are not related in general to isomorphic graphs. In order to ensure an affirmative statement in the last regard, this paper deals with a new proposal to establish a relationship between evolution algebras and graph theory. This is based on a recent work of the authors [7], who identify any algebra over a finite field with a pair of vertex-coloured graphs so that every isomorphism and every isotopism of the algebra under consideration gives rise to an isomorphism of the corresponding graph. We focus in particular on the study of isotopisms because of their importance to formulate algebraically the mutation of genotypes in the inheritance process [2]. Currently, it is known [5, 6] the distribution into isotopism classes of twoand three-dimensional evolution algebras over any base field, which determines in turn the spectrum of genetic patterns of two and three distinct genotypes during a mitosis process. 2 Mutation graphs Let Abe an algebra over a finite field Kand let Ann(A) = {u∈A|uv = 0,for all v∈A}denote its annihilator. Let us describe the step-by-step construction of the mutation graph of the algebra A. In order to illustrate this construction, we refer the reader to Figure 1.1, where we describe in detail how to obtain the mutation graph of the evolution algebra over F2with genetic pattern (e1, e1). Further, Figure 1.2 shows the spectrum of mutation graphs related to non-trivial genetic patterns of two distinct genotypes during a mitosis process. •Step 1: According to the description proposed by the authors in [7], we define the vertex-coloured graph G1(A) with the following four maximal monochromatic subsets of vertices RA={ru|u∈A\Ann(A)}, CA={cu|u∈A\Ann(A)}, SA={su|u∈A2\ {0}} and TA={tu,v |u, v ∈A, uv 6= 0}, and set of edges {rutu,v, cvtu,v, swtu,v |u, v, w ∈A, uv =w6= 0}. Suppose the vertices of the four sets RA,CA,SAand TAto be respectively coloured with the colours red, blue, green and black. •Step 2: We construct the edge-coloured line graph G2(A) associated to G1(A). The colours of its edges are inherited in natural way from those of the corresponding vertices in G1(A). Let u, v, w ∈Abe such that uv =w6= 0. Each triple (rutu,v, cvtu,v, swtu,v) of edges in G1(A) gives rise to a triangle in G2(A), which we call structural triangle in G2(A). Its edges are called structural, whereas the rest of edges in G2(A) are called non-structural. 3 Lemma 1. The following results hold. a) Every structural edge in G2(A)is coloured in black. b) The colours of every pair of non-structural edges that are incident to the same vertex in G2(A)coincide. c) The colours of every pair of non-structural edges that are respectively incident to a pair of distinct vertices of the same structural triangle in G2(A)are distinct. •Step 3: We define the edge-coloured multigraph G3(A) that results after contracting the three vertices of each structural triangle in G2(A). Proposition 1. The following results hold. a) The colours of every pair of parallel edges in G3(A)are distinct. b) There do not exist three parallel edges in G3(A). c) There always exists a green edge in any pair of parallel edges in G3(A). •Step 4: We define the edge-coloured graph G4(A) that results after contracting every pair of parallel edges in G3(A). The contraction of a pair of red-green parallel edges gives rise to a pink edge, whereas that related to a pair of blue-green parallel edges gives rise to an orange edge. Step 1: Vertex-coloured graph. Step 2: Line graph. Step 3: Contracted multigraph. Step 4: Mutation graph. Figure 1.1: Step-by-step construction of the mutation graph related to the bidimensional evolution algebra over F2of genetic pattern (e1, e2). Theorem 1. If two evolution algebras over a finite field are isomorphic, then their mutation graphs are isomorphic. 4Mutation graphs of asexual diploid organisms (e1,0) (e1, e1) (e1, e2) Figure 1.2: Spectrum of non-trivial mutation graphs for two distinct genotypes. References [1] Cabrera Casado, Y., Siles Molina, M. and Velasco, M. V. Evolution algebras of arbitrary dimension and their decompositions, Linear Algebra App.,495 (2016), 122–162. [2] Campos, T. M. M. and Holgate, P. Algebraic Isotopy in Genetics, IMA J. Math. Appl. Med. Biol. 4(1987) 215–222. [3] Elduque, A. and Labra, A. Evolution algebras and graphs, J. Algebra Appl., 14 (2015), 1550103, 10 pp. [4] Elduque, A. and Labra, A. On nilpotent evolution algebras, J. Linear Algebra Appl.,505 (2016), 11–31. [5] Falc´on, O. J., Falc´on, R. M. and N´u˜nez, J. Classification of asexual diploid organisms by means of strongly isotopic evolution algebras defined over any field, J. Algebra,472 (2017), 573–593. [6] Falc´on, O. J., Falc´on, R. M. and N´u˜nez, J. Algebraic computation of genetic patterns related to three-dimensional evolution algebras. Preprint. [7] Falc´on, O. J., Falc´on, R. M., N´u˜nez, J., Pacheco, A. M. and Villar, M. T. Computation of isotopisms of algebras over finite fields by means of graph invariants, J. Comput. Appl. Math.,318 (2017), 307–315. [8] Tian, J. P. Evolution Algebras and their Applications, Lect. Notes Math., 1921, Springer-Verlag, Berlin, 2008. [9] Tian, J. P. and Vojtechovsky, P. Mathematical concepts of evolution algebras in non-mendelian genetics, Quasigroups Related Systems,14 (2006), 111– 122. View publication statsView publication stats